To analyze complexity growth [1]}[2]} I will use the discrete random circuit model with q-local gates.
The complexity growth is governed by the number of gates in each step of the quantum circuit, and the complexity of the gates themselves. The number of q-local gates in a single step is \(N/q.\)  The complexity of a random \(q\) -local gate is the number of two-qubit gates that it takes to synthesize the \(q\) -qubit gate. For almost all \(q\) -qubit gates the complexity isFor a system of \(q\)  qubits, the maximum complexity of a unitary operator is \(4^q.\)  Furthermore, almost all operators are close to maximally complex. See [2]} and Brown, Freedman, Lin, Susskind: to appear.,
\({\cal {C}}_q = 4^q.\) 
